# How to evaluate a Riemann (Darboux?) integral?

This is the definition of integrable we have :

A bounded function $f : [a,b] \to \mathbb {R}$ is said to be integrable if $\sup\, L(f,P)= \inf\, U(f,P)$.

I've looked through the book but it involves the 'norm' of a partition, which we sort of don't have..

My lecturer gave an example which involved ( I think ) the uniform partition and creating a 'sequence' of partitions. I don't remember much more, so could someone help me out?

Edited the title as everywhere I look for Riemann integrals, the norm shows up, and the Darboux one seems to be the one we have, but my lecturer has always called it Riemann integral, so I'm not sure

• You have the $\sup, \inf$ mixed up there. – zhw. Jan 14 '18 at 8:44
• A Riemann integral can be defined as $\sup/\inf$ of Darboux sums as well as a limit of Riemann sums as norm of partition tends to $0$. That these two definitions are equivalent is a non-obvious fact which is somewhat difficult to prove. Moreover none of these definitions are used in evaluating integrals. Integrals are almost always evaluated using theorems meant to evaluate them and these theorems in turn are proved using these definitions. – Paramanand Singh Jan 14 '18 at 9:15
• Also fixed minor typo in the use of $\sup, \inf$ in the definition. – Paramanand Singh Jan 14 '18 at 9:19
• But how do we evaluate them using definitions? For example, the integral of $x^3$ from 0 to 1? – Saad Jan 14 '18 at 9:24
• Ok form a partition $P$ of $[0,1]$ into $n$ intervals via points $x_i=i/n$ and find $L(f, P), U(f, P)$ and use the non-trivial theorem that sup, inf of these numbers is same as the limit of these numbers when norm of partition (here $1/n$) tends to $0$. – Paramanand Singh Jan 14 '18 at 9:41